An invariant. It is also proved in an elementary geometric way that (in our notation) the function
\((1-c^2)xv_1+yv_2\)  is invariant for the billiard map (i.e. is an integral) for \(v=(v_1,v_2)\) , and indeed its square is found to be \((1-c^2)(1-s)\) . This corresponds to the constancy of a certain (easily written down) rational function on the above elliptic curves. We also note that there is an area form, invariant for the billiard map, on the phase space, given by \(\sin \alpha \cdot {\rm d}\alpha \wedge {\rm d}t\) , where \(t\)  is an arc-length parameter on \(C\)  and, for \((p,v)\)  in the phase-space, \(\alpha \)  is the angle between \(v\)  and the positive tangent to \(C\)  at \(p\)  (this was discovered by Birkhoff [1]}, see also [2]}, p. 33).
